Question:A ball is launched upward from a platform, and its height \(\mathrm{h(t)}\), in meters, t seconds after launch is modeled...

1. TRANSLATE the problem information

• Given information:
  • Height function: \(\mathrm{h(t) = -4.9t^2 + 19.6t + 3}\)
  • Need to find: time when ball reaches maximum height
• This is asking for when a quadratic function reaches its maximum value

2. INFER the mathematical approach

• Since the coefficient of \(\mathrm{t^2}\) is negative (-4.9), this parabola opens downward
• A downward-opening parabola has its maximum at the vertex
• For any quadratic \(\mathrm{f(t) = at^2 + bt + c}\), the vertex occurs at \(\mathrm{t = -b/(2a)}\)

3. SIMPLIFY using the vertex formula

• Identify the coefficients: \(\mathrm{a = -4.9, b = 19.6, c = 3}\)
• Apply the vertex formula: \(\mathrm{t = -b/(2a)}\)
• Substitute: \(\mathrm{t = -19.6/(2(-4.9))}\) \(\mathrm{= -19.6/(-9.8)}\) \(\mathrm{= 19.6/9.8}\) \(\mathrm{= 2}\)

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing this as a vertex-finding problem for a quadratic function.

Students may see the quadratic equation but not immediately connect that 'maximum height' means finding the vertex. They might try to set \(\mathrm{h(t) = 0}\) to find when the ball hits the ground, or attempt other approaches that don't directly address the maximum.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when applying the vertex formula.

Students correctly identify the need to use \(\mathrm{t = -b/(2a)}\) but make computational mistakes. Common errors include:

• Wrong signs: \(\mathrm{t = -19.6/(2(4.9))}\) instead of \(\mathrm{t = -19.6/(2(-4.9))}\)
• Arithmetic mistakes: Getting \(\mathrm{19.6/9.8 \neq 2}\)
• Not simplifying the fraction properly

This may lead them to select incorrect numerical answers if given multiple choice options.

The Bottom Line:

This problem tests whether students can connect the physical concept of 'maximum height' with the mathematical concept of finding a parabola's vertex, then execute the vertex formula correctly.